System of Particles and Rotational Motion Test

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System of Particles and Rotational Motion Test - 41

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Weekly Quiz Competition

Question 1
1 / -0

A gun of mass $$M$$, frees a shell of mass $$m$$ horizontally and the energy of explosion is such as would be sufficient to project the shell vertically to a height '$$h$$'. The recoil velocity of the gun is:

A
$$\left(\dfrac{2m^2gh}{M(m + M)}\right)^{\dfrac{1}{2}}$$

B
$$\left(\dfrac{2m^2gh}{M(m - M)}\right)^{\dfrac{1}{2}}$$

C
$$\left(\dfrac{2m^2gh}{2M(m - M)}\right)^{\dfrac{1}{2}}$$

D
$$\left(\dfrac{2m^2gh}{2M(m + M)}\right)^{\dfrac{1}{2}}$$

Solution

Applying linear momentum conservation,
$$p_i=p_f$$
$$\Rightarrow 0=mV_g=mV_b$$
$$\Rightarrow V_b=\dfrac{mV_g}{m}$$......(1)
According to question, total explosion energy is equal to $$mgh$$
$$\Rightarrow \dfrac{1}{2}mV_g^2+\dfrac{1}{2}mV_b^2=mgh$$
$$mv_g^2+m\left(\dfrac{M}{m}\right)^2V62g=2mgh$$ ( By (1))
$$v_g^2\left(M+\dfrac{M^2}{m}\right)=2mgh$$
$$v_g^2=2mgh\times \dfrac{m}{m(m+M)}$$
$$v_g=\sqrt{\dfrac{2m^2gh}{M(m+M)}}$$ (A)
Question 2
1 / -0

The ratio of the radii of gyration of a hollow sphere and a solid sphere of the same masses and radii about an axis passing through their centre is

A
$$\sqrt{2/3}$$

B
$$\sqrt{2/5}$$

C
$$\sqrt{5/3}$$

D
$$\sqrt{5/2}$$

Solution

The radius of gyration of a solid sphere is $$\sqrt{2/5}$$ and that of a hollow sphere is $$\sqrt{2/3}$$. Dividing one by the other, we get, the ratio = $$\sqrt{5/3}$$
Question 3
1 / -0

A constant torque is applied to a rigid body, whose moment of inertia is $$4 kg-m^2$$ around the axis of rotation. If the wheel starts from rest and attains an angular velocity of 20 rad/s in 10 s, what is the applied torque

A
$$18 N-m$$

B
$$8 N-m$$

C
$$4 N-m$$

D
$$6 N-m$$

Solution

Torque = $$I \alpha=4(20-0)/10=8 N-m$$

The correct option is (b)
Question 4
1 / -0

A ring of mass m and radius r rolls on an inclined plane. A torque is produced in the ring due to

A
$$mg sin \theta$$ and the radius of the ring

B
the perpendicular distance between the force and the point of contact and the force $$mg$$ acting on it

C
the perpendicular distance between the force and the point of contact and the normal force acting on the object

D
the perpendicular distance between the force and the point of contact and the force $$mg sin \theta$$ acting on it

Solution

Torque is determined using the formula
$$r\times F=rF\sin\theta$$
Here $$r\sin\theta$$ stands for perpendicular distance between the force and the point of contact as shown in figure
Question 5
1 / -0

Three objects of same mass but different geometries, capable of rotating have radius of gyration as 0.2, 0.5 and 0.7. A torque is applied to these objects when they are rotating with constant angular velocity. Which object will have a larger response time for showing the change in their angular velocity

A
Object with a radius of gyration of 0.7 will take a long time to respond to the torque

B
Object with a radius of gyration of 0.5 will take a long time to respond to the torque

C
Object with a radius of gyration of 0.2 will take a long time to respond to the torque

D
All objects with have same response time

Solution

The property of moment of inertia I is a measure of rotational inertia of the body. Larger the moment of inertia, larger the object resists to the sudden increase or decrease of the speed. It allows a gradual change in the speed and prevents jerky motions.
Question 6
1 / -0

Statement related to center of gravity that is incorrect is

A
The center of gravity of an object is defined as the point through which its whole weight appears to act.

B
The center of gravity is sometimes confused with the center of mass.

C
The center of gravity always lies inside the object.

D
For an object placed in a uniform gravitational field, the center of gravity coincides with the center of mass.

Solution

In case of a ring, the centre of gravity lies at the centre of the ring. Hence option c is the incorrect option
Question 7
1 / -0

In the image shown . AC is a rigid rod of length 12 cms rotating about point P, whose velocities are shown in the figure. Find the distance PC = x

A
12 cms

B
8 cms

C
4 cms

D
2 cms

Solution

Since the rod is rotating about the point P, the angular velocities of all the points are same.

Thus, the linear velocities of point C and A are related to the angular velocity $$\omega $$ as $$x \omega=3$$ and $$(d+x)\omega$$=12.

Solving both the equations, we get $$x=4cm$$
Question 8
1 / -0

Which of the below statement is correct

A
All points in an object move in a straight line in rectilinear motion, while moves in a curved path when rotated about a fixed point

B
All points in an object move with the same linear speed when rotated about a fixed point

C
All points in an object move in a straight line, while the object moves in a curved path pivoted about a fixed point

D
All points in an object moves in a curved path, while the object is moving along a straight line

Solution

In a rectilinear motion, all points in an object move in a straight line
In a circular motion, all points moved in a curved path, if the object is pivoted about a point
Question 9
1 / -0

A jar filled with two non mixing liquids $$1$$ and $$2$$ having densities $$\rho_{1}$$ and $$\rho_{2}$$ respectively. A solid ball, made of a material of density $$\rho_{3}$$, is dropped in the jar. It comes to equilibrium in the position shown in the figure

A
$$\rho_{3} < \rho_{1} < \rho_{2}$$

B
$$\rho_{1} < \rho_{3} < \rho_{2}$$

C
$$\rho_{1} < \rho_{2} < \rho_{3}$$

D
$$\rho_{3} < \rho_{2} < \rho_{1}$$

Solution

Basic principal of density is that higher density material settle down and lower density floats. So here we can see that
$$\rho_1<\rho_3<\rho_2$$
$$\rho_3$$ density of solid base
$$\rho_1 \&\rho_2$$ density of liquids
Question 10
1 / -0

A wheel having moment of inertia $$4\ kg\ {m}^{2}$$ about its symmetrical axis, rotates at rate of $$240$$ rpm about it. The torque which can stop the rotation of the wheel in one minute is:

A
$$\dfrac {5\pi}{7}Nm$$

B
$$\dfrac {8\pi}{15}Nm$$

C
$$\dfrac {2\pi}{9}Nm$$

D
$$\dfrac {3\pi}{7}Nm$$

Solution

$$\alpha =\dfrac{240\times 2\pi}{60\times (60)}$$
$$\alpha =\dfrac{8\pi}{60}$$
now $$T=I\alpha$$
$$=4\times \dfrac{8\pi}{60}$$
$$=\dfrac{8\pi}{15}$$

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System of Particles and Rotational Motion Test - 41

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System of Particles and Rotational Motion Test - 41

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Question 1

$$\left(\dfrac{2m^2gh}{M(m + M)}\right)^{\dfrac{1}{2}}$$

$$\left(\dfrac{2m^2gh}{M(m - M)}\right)^{\dfrac{1}{2}}$$

$$\left(\dfrac{2m^2gh}{2M(m - M)}\right)^{\dfrac{1}{2}}$$

$$\left(\dfrac{2m^2gh}{2M(m + M)}\right)^{\dfrac{1}{2}}$$

Solution

Question 2

$$\sqrt{2/3}$$

$$\sqrt{2/5}$$

$$\sqrt{5/3}$$

$$\sqrt{5/2}$$

Solution

Question 3

$$18 N-m$$

$$8 N-m$$

$$4 N-m$$

$$6 N-m$$

Solution

Question 4

$$mg sin \theta$$ and the radius of the ring

the perpendicular distance between the force and the point of contact and the force $$mg$$ acting on it

the perpendicular distance between the force and the point of contact and the normal force acting on the object

the perpendicular distance between the force and the point of contact and the force $$mg sin \theta$$ acting on it

Solution

Question 5

Object with a radius of gyration of 0.7 will take a long time to respond to the torque

Object with a radius of gyration of 0.5 will take a long time to respond to the torque

Object with a radius of gyration of 0.2 will take a long time to respond to the torque

All objects with have same response time

Solution

Question 6

The center of gravity of an object is defined as the point through which its whole weight appears to act.

The center of gravity is sometimes confused with the center of mass.

The center of gravity always lies inside the object.

For an object placed in a uniform gravitational field, the center of gravity coincides with the center of mass.

Solution

Question 7

12 cms

8 cms

4 cms

2 cms

Solution

Question 8

All points in an object move in a straight line in rectilinear motion, while moves in a curved path when rotated about a fixed point

All points in an object move with the same linear speed when rotated about a fixed point

All points in an object move in a straight line, while the object moves in a curved path pivoted about a fixed point

All points in an object moves in a curved path, while the object is moving along a straight line

Solution

Question 9

$$\rho_{3} < \rho_{1} < \rho_{2}$$

$$\rho_{1} < \rho_{3} < \rho_{2}$$

$$\rho_{1} < \rho_{2} < \rho_{3}$$

$$\rho_{3} < \rho_{2} < \rho_{1}$$

Solution

Question 10

$$\dfrac {5\pi}{7}Nm$$

$$\dfrac {8\pi}{15}Nm$$

$$\dfrac {2\pi}{9}Nm$$

$$\dfrac {3\pi}{7}Nm$$

Solution

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